Section 4.9Laplace’s equation in cylindrical coordinatesAs in the case of spherical coordinates, this equation is solved by a series expansion in terms of products of functions of theindividual cylindrical coordinates. That is, we use separation of variables.Φ(ρ, ϕ, z)=Xl,m,nαlmnUl(ρ)Vm(ϕ)Wk(z).(4.53)SubstitutingΦlmk(ρ, ϕ, zUl(ρ)Vm(ϕ)Wk(z).into Eq 4.52 and dividing byΦlmk(ρ, ϕ, z), we can sequentiallyseparate all the variable .dependence:1Ulρddρ·ρddρUl¸+1Vmρ2d2dϕ2Vm=−1Wnd2dz2Wk=±k2(4.54)ρUlddρ·ρddρUl¸∓ρ2k2=−1Vmd2dϕ2Vm=±m2(4.55)These functions satisfy the Eqs. 4.54, 4.55 and 4.56:d2Wk(z)dz2=λzWk(z);λz=k2(4.54a)Wk(zAkekz+Bke−kz(4.54b)Note thatkcan be complex, pure imaginary or real.d2dϕ2Vm(ϕλϕVm(ϕ);λϕ=−m2(4.55)Vm(ϕCmeimϕ+Dme−imϕ(4.55b)Note thatmcould be non-integer and/or complexρ2d2dρ2U(ρ)+ρddρU(ρ¡k2ρ2−m2¢U(ρ)=0(4.56)(kρ)2d2d()2Um(dd()Um(¡()2−m2¢Um(Um(amJm(bmNm()(4.56b)wherenowthelabelonUm()becomesmand theJmand .Nmare called Bessel functions of the
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This note was uploaded on 12/19/2010 for the course PHYS 411 taught by Professor G during the Spring '10 term at Missouri S&T.